Exercises

(1)
One of the initial value problems below has more than one solution. Which one is it?
(a)
\(y^{\,\prime }=\dfrac {x}{y+1}\), \(y(0)=0\)
(b)
\(xy^{\,\prime }=\ln {(y)}\), \(y(1)=e\)
(c)
\(xy^{\,\prime }=\ln {(y)}\), \(y(1)=1\)
(d)
\(y^{\,\prime }=xy^{3/4}\), \(y(0)=1\)
(e)
\(y^{\,\prime }=xy^{3/4}\), \(y(0)=0\)
(f)
\(y^{\,\prime }=xy^{4/3}\), \(y(0)=1\)
(g)
\(y^{\,\prime }=xy^{4/3}\), \(y(0)=0\)
(2)
Consider the IVP \(y^{\,\prime }=\dfrac {xy^{1/3}}{y+1}\), \(y(0)=a\) where \(a\) is a constant. For how many values of \(a\) does the Existence and Uniqueness Theorem NOT guarantee the existence of a unique solution to the IVP (on some open interval)?
(3)
Suppose \(y(x)\) satisfies \(\displaystyle y^{\,\prime }=y-y\ln {\left (y\right )}\), \(y(1)=e\). Find \(y(2026)\) to the nearest hundredth.
(4)
Let \(a\) be a constant. Suppose \(\displaystyle y^{\,\prime }=\frac {\left (y^5-y^3\right )^{1/3}}{1-e^{y^2}}\), \(y(0)=a\). Determine a value of \(a\) for which the Existence and Uniquesss Theorem guarantees that a solution exists to this IVP, but where it does not guarantee a unique solution (on some open interval).
(5)
Suppose \(y^{\,\prime }=f(x,y)\) has the slope field below. There is exactly one value of \(0.5<a<5\) for which it is not guaranteed that the IVP \(y^{\,\prime }=f(x,y)\), \(y(0)=a\) has a unique solution. Determine \(a\) to the nearest tenth.
A direction field. [Picture]
Figure 1: A direction field (also known as a slope field)